Rough Topology Research Articles

Neutrosophic Soft Rough Topology and its Applications to Multi-Criteria Decision-Making

Neutrosophic Soft Rough Topology and its Applications to Multi-Criteria Decision-Making

A multi-scale FEM-BEM formulation for contact mechanics between rough surfaces

A novel multi-scale finite element formulation for contact mechanics between nominally smooth but microscopically rough surfaces is herein proposed. The approach integrates the interface finite element method (FEM) for modelling interface interactions at the macro-scale with a boundary element method (BEM) for the solution of the contact problem at the micro-scale. The BEM is used at each integration point to determine the normal contact traction and the normal contact stiffness, allowing to take into account any desirable kind of rough topology, either real, e.g. obtained from profilometric data, or artificial, evaluated with the most suitable numerical or analytical approach. Different numerical strategies to accelerate coupling between FEM and BEM are discussed in relation to a selected benchmark test.

On modified soft rough sets on a complete atomic Boolean lattice

Uncertainty has been used in different areas in our life such as medical diagnosis, social studies, pharmacology and others. The soft set theory and rough set theory are treated as mathematical approaches to deal with uncertainty. In 2011, Feng et al. introduced the notion of soft rough sets. In 2013, we introduced the notion of soft rough sets on a complete atomic Boolean lattice as a generalization of soft rough sets. In this paper, we strengthen the concept of soft rough set on a complete atomic Boolean lattice by defining the concept of modified soft rough set (MRS-set) on a complete atomic Boolean lattice. In this model, some properties which were not satisfied in soft rough sets on a complete atomic Boolean lattice can be proved. Finally, we will use this concept to introduce the concept of modified soft rough topology and we apply this concept in diabetes mellitus.

On Soft Rough Topology with Multi-Attribute Group Decision Making

On Soft Rough Topology with Multi-Attribute Group Decision Making

Rough Spaces on Covering Based Rough Sets

In this paper, we discuss open (closed) sets, rough interiors, rough closures, continuous mappings, open (closed) mappings and homeomorphism mappings of covering based rough topology of Akduman, Özcelik and Özel. We also construct the nano topology on a given covering based rough set.

Novel concepts of soft rough set topology with applications

Novel concepts of soft rough set topology with applications

On Soft Rough Topology with Multi-Attribute Group Decision Making

Rough set approaches encounter uncertainty by means of boundary regions instead of membership values. In this paper, we develop the topological structure on soft rough set ( SR -set) by using pairwise SR -approximations. We define SR -open set, SR -closed sets, SR -closure, SR -interior, SR -neighborhood, SR -bases, product topology on SR -sets, continuous mapping, and compactness in soft rough topological space ( SRTS ). The developments of the theory on SR -set and SR -topology exhibit not only an important theoretical value but also represent significant applications of SR -sets. We applied an algorithm based on SR -set to multi-attribute group decision making (MAGDM) to deal with uncertainty.

Covering-Based Rough Fuzzy, Intuitionistic Fuzzy and Neutrosophic Nano Topology and Applications

In recent years, “mathematical orientations on real-life problems”, which continue to increase, began to make a significant impact. Information systems for many decision-making problems consist of uncertain, incomplete, indeterminate and indiscernible structures and components. Classical set theory and interpretation methods fail to represent, express and solve the problems of these types or cause to make wrong decisions. For this reason, in this study, we provide definitions and methods to present information and problem representations in more detail and precision. This paper introduces three new topologies called covering-based rough fuzzy, covering-based rough intuitionistic fuzzy and covering-based rough neutrosophic nano topology. Some fundamental definitions such as open set, closed set, interior, closure and basis are given. Neutrosophic definitions and properties are mainly investigated. We give some real life applications of covering-based rough neutrosophic nano topology in the final part of the paper and an explanatory example of decision making application by defining core point.

On Multi-Fuzzy Rough Sets, Relations, and Topology

This article describes how rough set theory has an innate topological structure characterized by the partitions. The approximation operators in rough set theory can be viewed as the topological operators namely interior and closure operators. Thus, topology plays a role in the theory of rough sets. This article makes an effort towards considering closed sets a primitive concept in defining multi-fuzzy topological spaces. It discusses the characterization of multi-fuzzy topology using closed multi-fuzzy sets. A set of axioms is proposed that characterizes the closure and interior of multi-fuzzy sets. It is proved that the set of all lower approximation of multi-fuzzy sets under a reflexive and transitive multi-fuzzy relation forms a multi-fuzzy topology.

Roughness measures of locally finite covering rough sets

The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC-, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC-space (U,C) and a nonempty set X(⊆U). Using a reduction of a given LFC-space (U,C), the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC-spaces (U,C), where each of the cardinalities of the sets U, C, and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC-spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X(⊆U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness.

Decision Making Using Rough Topology and Indiscernibility Matrix for Diagnosing Disease

Decision Making Using Rough Topology and Indiscernibility Matrix for Diagnosing Disease

Silicone/graphene oxide sheet-alumina nanorod ternary composite for superhydrophobic antifouling coating

Superhydrophobic silicone fouling-release (FR) nanocomposite coatings have gained considerable interest as an eco-friendly and economic solution to combat biofouling problems of ship hulls. In this work, graphene oxide anchored with alumina nanorods (GO-Al2O3 NRs) sheet hybrids were fabricated via a two-phase method. A ternary nanocomposite series of elastomeric silicone/GO-Al2O3 hybrid NRs sheet coating was developed via solution-casting method. γ–Al2O3 NRs of 150 nm in length and 20 nm in diameter were fastened between GO nanosheets and conformably coated with silicone layers. Different concentrations of GO-Al2O3 hybrid NRs sheet fillers were inserted in the silicone composites to study the structure property relationship. Nanocomposites’ surface characteristics, such as superhydrophobicity, surface free energy (SFE), and rough topology were investigated here. The physical and mechanical characteristics of the coating materials were also considered. Selected micro-foulants were used for the investigation of biological inertness. Laboratory assessments were performed on coated specimens for up to 28 days. A rigorous field trial was carried out for the fabricated model in natural marine water for 3 months. Well-dispersed GO-γ-Al2O3 NRs sheets hybrids of up to 1 wt.% increased the contact angle (151°), decreased the SFE (13.25), and provided micro-nano roughness. Thus, a promising FR nanocomposite coating was approved with superhydrophobicity, high thermal stability, and economic savings for marine coating applications.

Low energy ion irradiation studies of fullerene C70 thin films – An emphasis on mapping the local structure modifications

Metal matrix composites have a diverse range of applications. We have probed local structural modifications taking fullerene C70 as a model matrix material. In this study, thin films of fullerene C70 were grown on polished silicon and quartz substrates by resistive heating of fullerene C70. In order to understand local structural changes, these semi-crystalline films were irradiated with 120 keV N+ ions at different fluences ranging from 1 × 1013 to 3 × 1016 ions.cm−2. Microscopic analysis shows an increase in topological roughness up to a fluence of 1 × 1015 ions.cm−2 and then a decrease. The size of the particles decreases with increase in the fluence. Various spectroscopic analyses conducted on differently irradiated fullerene C70 films show that the bandgap decreases with the increase in the fluence which will increase the conductivity of the irradiated thin films. These results are proved by I-V measurements showing the decrease in the resistivity at higher fluences. Therefore, this characteristic trait makes fullerene C70 to be used as a prominent matrix component in the composites. Raman spectroscopic investigations reveal that C70 is completely transformed into amorphous carbon at a fluence of 3 × 1016 ions.cm−2. Different vibrational modes in FT-IR are also reported in the present work.

A support and density theorem for Markovian rough paths

We establish two results concerning a class of geometric rough paths $\mathbf{X}$ which arise as Markov processes associated to uniformly subelliptic Dirichlet forms. The first is a support theorem for $\mathbf{X}$ in $\alpha$-H\"older rough path topology for all $\alpha \in (0,1/2)$, which answers in the positive a conjecture of Friz-Victoir (2010). The second is a H\"ormander-type theorem for the existence of a density of a rough differential equation driven by $\mathbf{X}$, the proof of which is based on analysis of (non-symmetric) Dirichlet forms on manifolds.

Lock-exchange release density currents over three-dimensional regular roughness elements

A fundamental study has been conducted to understand the front characteristics and the mixing in the flow of density currents over rough surfaces. A large-eddy simulation (LES) has been performed for lock-exchange release density currents over rough walls to shed light on the unsteady mixing processes. A volume-penalization method, which is a special case of the immersed-boundary method, has been implemented to realize the bottom-mounted rough topology. In this study, the LES has been conducted in a channel with a lower wall covered with three-dimensional cube- and pyramid-shaped roughness elements, such that all cases have the same base area, but differences in the roughness solidity and volume fraction of roughness. Both cases of identical roughness elements and those with randomness in height have been considered. The maximum roughness height for all cases is kept at a constant fraction (10 %) of the total channel height. The study focuses on the instantaneous mixing processes in lock-exchange release currents over rough surfaces. An important contribution of the work is that qualitative and quantitative analysis has been conducted to demonstrate additional mixing mechanisms due to the presence of surface roughness that enhances dilution of the current. Enhanced mixing due to roughness is related to the strength of the shear layer resulting from the roughness, and hence depends on friction Reynolds number ($Re_{\unicode[STIX]{x1D70F}}$). The combined role of current characteristics and $Re_{\unicode[STIX]{x1D70F}}$ together dictate the mixing processes and extent of dilution in density currents over surface roughness.

Distributed Roughness Effects on Transitional and Turbulent Boundary Layers

A numerical investigation is carried out to study the transition of a subsonic boundary layer on a flat plate with roughness elements distributed over the entire surface. Post-transition, the effect of surface roughness on a spatially developing turbulent boundary layer (TBL) is explored. In the transitional regime, the onset of flow transition predicted by the current simulations is in agreement with the experimentally based correlations proposed in the literature. Transition mechanisms are shown to change significantly with the increasing roughness height. Roughness elements that are inside the boundary layer create an elevated shear layer and alternating high and low speed streaks near the wall. Secondary sinuous instabilities on the streaks destabilize the shear layer promoting transition to turbulence. For the roughness topology considered, it is observed that the instability wavelengths are governed by the streamwise and spanwise spacing between the roughness elements. In contrast, the roughness elements that are higher than the boundary layer create turbulent wakes in their lee. The scale of instability is much shorter and transition occurs due to the shedding from the obstacles. Post-transition, in the spatially developing TBL, the velocity defect profiles for both the smooth and rough walls collapsed when non dimensionalized in the outer units. However, when compared to the smooth wall, deviation in the Reynolds stresses are observable in the outer layer; the deviation being higher for the larger roughness elements.

ROUGH TOPOLOGY ON APPROXIMATION SPACES

Though there are many studies on the topological properties of rough set approximations, only a few works have been done on the concept of rough topology. In this paper, a new definition of rough topology on an approximation space is proposed using the rough subsets of the extended approximation space. The basic concepts of a topological space are extended to the proposed rough topological space and the properties are investigated.

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

A careful look at rough path topology applied to Brownian motion reveals new possible properties of the well-known Lévy area, in particular the presence of an intrinsic drift of this area. Using renormalization limit of Markov chains on periodic graphs, we present a construction of such a non-trivial drift and give an explicit formula for it. Several examples with explicit computations are included.

Electrohydrodynamic spinning of random-textured silver webs for electrodes embedded in flexible organic solar cells

A convenient process for fabricating a transparent conducting electrode on a flexible substrate is essential for numerous low-cost optoelectronic devices, including organic solar cells (OSCs), touch sensors, and free-form lighting applications. Solution-processed metal-nanowire arrays are attractive due to their low sheet resistance and optical clarity. However, the limited conductance at wire junctions and the rough surface topology still need improvement. Here, we present a facile process of electrohydrodynamic spinning using a silver (Ag) - polymer composite paste with high viscosity. Unlike the metal-nanofiber web formed by conventional electrospinning, a relatively thick, but still invisible-to-naked eye, Ag-web random pattern was formed on a glass substrate. The process parameters such as the nozzle diameter, voltage, flow rate, standoff height, and nozzle-scanning speed, were systematically engineered. The formed random texture Ag webs were embedded in a flexible substrate by in-situ photo-polymerization, release from the glass substrate, and post-annealing. OSCs with a donor-acceptor polymeric heterojunction photoactive layer were prepared on the Ag-web-embedded flexible films with various Ag-web densities. The short-circuit current and the power conversion efficiency of an OSC with a Ag-web-embedded electrode were not as high as those of the control sample with an indium-tin-oxide electrode. However, the Ag-web textures embedded in the OSC served well as electrodes when bent (6-mm radius), showing a power conversion efficiency of 2.06% (2.72% for the flat OSC), and the electrical stability of the Ag-web-textured patterns was maintained for up to 1,000 cycles of bending.

Application of nanomechanical response of wrinkled thin films in surface feature generation

Wrinkle patterns on the elastomeric substrate are gaining significant attention these days because of its wide variety of applications in the area of stretchable electronics, bacterial adhesion, surface modifications, etc. Development of controlled patterns on these wrinkled thin film surfaces can lead to different possible outcomes such as surface area enhancement, generation of rough surface topology, formation of template for nano-pillars, etc. which can further diversify and strengthen the application of wrinkled thin films. On a wrinkled surface, due to varying stress distribution and material plasticity along the slope, the indentation responses are expected to be a function of the position of indentation on the slope. In this work, this property is applied to generate controlled patterns on wrinkled polymeric thin film using nanoindentation (NI) technique. Synthesis of polymeric nano-pillars of uniform and varying heights and generation of enhanced surface area on flat and wrinkled surfaces are discussed here. Besides the inherent mechanical properties of a film material, the mechanical response of a thin film to indentation also depends on its thickness and the interfacial strength of adhesion with the substrate. These responses are studied for different thicknesses of wrinkled film with an objective to determine the necessary indentation characteristics for achieving target surface area enhancement and required nano-pillar heights. The height and the height distribution of these pillars can be varied by varying the film thickness, the applied indentation load, and the location of indentation.